论文标题
U级增长和较高常数的均等均值估计器的U统计量
U-statistics of growing order and sub-Gaussian mean estimators with sharp constants
论文作者
论文摘要
本文解决了以下问题：给定I.I.D.的样本。具有有限差异的随机变量，一个人是否可以构造一个未知平均值的估计器，其执行几乎和数据的分布是否正态分布？实现这一目标的最受欢迎的例子之一是估计器的中位数。但是，从结果界限中的常数次优效率是低效率的。我们表明，如果基本分布拥有超过$ \ frac {3+ \ sqrt {5}}} {2} {2} {2} \ ables to lorbebesgue，则对估算器的中位数进行了不变的修改，该偏差保证达最高$ 1+o（1）$。对于依赖估计器中位数作为基础的各种算法的算法，这一结果可取得潜在的改进。我们论点的核心是允许随着样本量增长的秩序统计量的新偏差不平等，这可能是独立的。
This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to $1+o(1)$ factor if the underlying distribution possesses more than $\frac{3+\sqrt{5}}{2}\approx 2.62$ moments and is absolutely continuous with respect to the Lebesgue measure. This result yields potential improvements for a variety of algorithms that rely on the median of means estimator as a building block. At the core of our argument is are the new deviation inequalities for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest.
